author  wenzelm 
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parent 47108  2a1953f0d20d 
child 50327  bbea2e82871c 
permissions  rwrr 
21164  1 
(* Title : Deriv.thy 
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Author : Jacques D. Fleuriot 

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Copyright : 1998 University of Cambridge 

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Conversion to Isar and new proofs by Lawrence C Paulson, 2004 

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GMVT by Benjamin Porter, 2005 

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*) 

7 

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header{* Differentiation *} 

9 

10 
theory Deriv 

29987  11 
imports Lim 
21164  12 
begin 
13 

22984  14 
text{*Standard Definitions*} 
21164  15 

16 
definition 

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deriv :: "['a::real_normed_field \<Rightarrow> 'a, 'a, 'a] \<Rightarrow> bool" 
21164  18 
{*Differentiation: D is derivative of function f at x*} 
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("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60) where 
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"DERIV f x :> D = ((%h. (f(x + h)  f x) / h)  0 > D)" 
21164  21 

22 
primrec 

34941  23 
Bolzano_bisect :: "(real \<times> real \<Rightarrow> bool) \<Rightarrow> real \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> real \<times> real" where 
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"Bolzano_bisect P a b 0 = (a, b)" 

25 
 "Bolzano_bisect P a b (Suc n) = 

26 
(let (x, y) = Bolzano_bisect P a b n 

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in if P (x, (x+y) / 2) then ((x+y)/2, y) 

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else (x, (x+y)/2))" 

21164  29 

30 

31 
subsection {* Derivatives *} 

32 

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lemma DERIV_iff: "(DERIV f x :> D) = ((%h. (f(x + h)  f(x))/h)  0 > D)" 
21164  34 
by (simp add: deriv_def) 
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lemma DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h)  f(x))/h)  0 > D" 
21164  37 
by (simp add: deriv_def) 
38 

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lemma DERIV_const [simp]: "DERIV (\<lambda>x. k) x :> 0" 

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by (simp add: deriv_def tendsto_const) 
21164  41 

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lemma DERIV_ident [simp]: "DERIV (\<lambda>x. x) x :> 1" 
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by (simp add: deriv_def tendsto_const cong: LIM_cong) 
21164  44 

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lemma DERIV_add: 

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"\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x + g x) x :> D + E" 

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by (simp only: deriv_def add_diff_add add_divide_distrib tendsto_add) 
21164  48 

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lemma DERIV_minus: 

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"DERIV f x :> D \<Longrightarrow> DERIV (\<lambda>x.  f x) x :>  D" 

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by (simp only: deriv_def minus_diff_minus divide_minus_left tendsto_minus) 
21164  52 

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lemma DERIV_diff: 

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"\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x  g x) x :> D  E" 

37887  55 
by (simp only: diff_minus DERIV_add DERIV_minus) 
21164  56 

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lemma DERIV_add_minus: 

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"\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x +  g x) x :> D +  E" 

59 
by (simp only: DERIV_add DERIV_minus) 

60 

61 
lemma DERIV_isCont: "DERIV f x :> D \<Longrightarrow> isCont f x" 

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proof (unfold isCont_iff) 

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assume "DERIV f x :> D" 

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hence "(\<lambda>h. (f(x+h)  f(x)) / h)  0 > D" 
21164  65 
by (rule DERIV_D) 
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hence "(\<lambda>h. (f(x+h)  f(x)) / h * h)  0 > D * 0" 
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by (intro tendsto_mult tendsto_ident_at) 
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hence "(\<lambda>h. (f(x+h)  f(x)) * (h / h))  0 > 0" 
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by simp 
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hence "(\<lambda>h. f(x+h)  f(x))  0 > 0" 
23398  71 
by (simp cong: LIM_cong) 
21164  72 
thus "(\<lambda>h. f(x+h))  0 > f(x)" 
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by (simp add: LIM_def dist_norm) 
21164  74 
qed 
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lemma DERIV_mult_lemma: 

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fixes a b c d :: "'a::real_field" 
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shows "(a * b  c * d) / h = a * ((b  d) / h) + ((a  c) / h) * d" 
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by (simp add: diff_minus add_divide_distrib [symmetric] ring_distribs) 
21164  80 

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lemma DERIV_mult': 

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assumes f: "DERIV f x :> D" 

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assumes g: "DERIV g x :> E" 

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shows "DERIV (\<lambda>x. f x * g x) x :> f x * E + D * g x" 

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proof (unfold deriv_def) 

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from f have "isCont f x" 

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by (rule DERIV_isCont) 

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hence "(\<lambda>h. f(x+h))  0 > f x" 

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by (simp only: isCont_iff) 

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hence "(\<lambda>h. f(x+h) * ((g(x+h)  g x) / h) + 
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((f(x+h)  f x) / h) * g x) 
21164  92 
 0 > f x * E + D * g x" 
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by (intro tendsto_intros DERIV_D f g) 
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thus "(\<lambda>h. (f(x+h) * g(x+h)  f x * g x) / h) 
21164  95 
 0 > f x * E + D * g x" 
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by (simp only: DERIV_mult_lemma) 

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qed 

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lemma DERIV_mult: 

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"[ DERIV f x :> Da; DERIV g x :> Db ] 

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==> DERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))" 

102 
by (drule (1) DERIV_mult', simp only: mult_commute add_commute) 

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104 
lemma DERIV_unique: 

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"[ DERIV f x :> D; DERIV f x :> E ] ==> D = E" 

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apply (simp add: deriv_def) 

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apply (blast intro: LIM_unique) 

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done 

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text{*Differentiation of finite sum*} 

111 

31880  112 
lemma DERIV_setsum: 
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assumes "finite S" 

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and "\<And> n. n \<in> S \<Longrightarrow> DERIV (%x. f x n) x :> (f' x n)" 

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shows "DERIV (%x. setsum (f x) S) x :> setsum (f' x) S" 

116 
using assms by induct (auto intro!: DERIV_add) 

117 

21164  118 
lemma DERIV_sumr [rule_format (no_asm)]: 
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"(\<forall>r. m \<le> r & r < (m + n) > DERIV (%x. f r x) x :> (f' r x)) 

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> DERIV (%x. \<Sum>n=m..<n::nat. f n x :: real) x :> (\<Sum>r=m..<n. f' r x)" 

31880  121 
by (auto intro: DERIV_setsum) 
21164  122 

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text{*Alternative definition for differentiability*} 

124 

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lemma DERIV_LIM_iff: 

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fixes f :: "'a::{real_normed_vector,inverse} \<Rightarrow> 'a" shows 
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"((%h. (f(a + h)  f(a)) / h)  0 > D) = 
21164  128 
((%x. (f(x)f(a)) / (xa))  a > D)" 
129 
apply (rule iffI) 

130 
apply (drule_tac k=" a" in LIM_offset) 

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apply (simp add: diff_minus) 

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apply (drule_tac k="a" in LIM_offset) 

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apply (simp add: add_commute) 

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done 

135 

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lemma DERIV_iff2: "(DERIV f x :> D) = ((%z. (f(z)  f(x)) / (zx))  x > D)" 
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by (simp add: deriv_def diff_minus [symmetric] DERIV_LIM_iff) 
21164  138 

139 
lemma DERIV_inverse_lemma: 

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"\<lbrakk>a \<noteq> 0; b \<noteq> (0::'a::real_normed_field)\<rbrakk> 
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\<Longrightarrow> (inverse a  inverse b) / h 
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=  (inverse a * ((a  b) / h) * inverse b)" 
21164  143 
by (simp add: inverse_diff_inverse) 
144 

145 
lemma DERIV_inverse': 

146 
assumes der: "DERIV f x :> D" 

147 
assumes neq: "f x \<noteq> 0" 

148 
shows "DERIV (\<lambda>x. inverse (f x)) x :>  (inverse (f x) * D * inverse (f x))" 

149 
(is "DERIV _ _ :> ?E") 

150 
proof (unfold DERIV_iff2) 

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from der have lim_f: "f  x > f x" 

152 
by (rule DERIV_isCont [unfolded isCont_def]) 

153 

154 
from neq have "0 < norm (f x)" by simp 

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with LIM_D [OF lim_f] obtain s 

156 
where s: "0 < s" 

157 
and less_fx: "\<And>z. \<lbrakk>z \<noteq> x; norm (z  x) < s\<rbrakk> 

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\<Longrightarrow> norm (f z  f x) < norm (f x)" 

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by fast 

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show "(\<lambda>z. (inverse (f z)  inverse (f x)) / (z  x))  x > ?E" 
21164  162 
proof (rule LIM_equal2 [OF s]) 
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fix z 
21164  164 
assume "z \<noteq> x" "norm (z  x) < s" 
165 
hence "norm (f z  f x) < norm (f x)" by (rule less_fx) 

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hence "f z \<noteq> 0" by auto 

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thus "(inverse (f z)  inverse (f x)) / (z  x) = 
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 (inverse (f z) * ((f z  f x) / (z  x)) * inverse (f x))" 
21164  169 
using neq by (rule DERIV_inverse_lemma) 
170 
next 

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from der have "(\<lambda>z. (f z  f x) / (z  x))  x > D" 
21164  172 
by (unfold DERIV_iff2) 
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thus "(\<lambda>z.  (inverse (f z) * ((f z  f x) / (z  x)) * inverse (f x))) 
21164  174 
 x > ?E" 
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by (intro tendsto_intros lim_f neq) 
21164  176 
qed 
177 
qed 

178 

179 
lemma DERIV_divide: 

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"\<lbrakk>DERIV f x :> D; DERIV g x :> E; g x \<noteq> 0\<rbrakk> 
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\<Longrightarrow> DERIV (\<lambda>x. f x / g x) x :> (D * g x  f x * E) / (g x * g x)" 
21164  182 
apply (subgoal_tac "f x *  (inverse (g x) * E * inverse (g x)) + 
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D * inverse (g x) = (D * g x  f x * E) / (g x * g x)") 

184 
apply (erule subst) 

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apply (unfold divide_inverse) 

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apply (erule DERIV_mult') 

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apply (erule (1) DERIV_inverse') 

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apply (simp add: ring_distribs nonzero_inverse_mult_distrib) 
21164  189 
done 
190 

191 
lemma DERIV_power_Suc: 

31017  192 
fixes f :: "'a \<Rightarrow> 'a::{real_normed_field}" 
21164  193 
assumes f: "DERIV f x :> D" 
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shows "DERIV (\<lambda>x. f x ^ Suc n) x :> (1 + of_nat n) * (D * f x ^ n)" 
21164  195 
proof (induct n) 
196 
case 0 

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show ?case by (simp add: f) 
21164  198 
case (Suc k) 
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from DERIV_mult' [OF f Suc] show ?case 

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apply (simp only: of_nat_Suc ring_distribs mult_1_left) 
29667  201 
apply (simp only: power_Suc algebra_simps) 
21164  202 
done 
203 
qed 

204 

205 
lemma DERIV_power: 

31017  206 
fixes f :: "'a \<Rightarrow> 'a::{real_normed_field}" 
21164  207 
assumes f: "DERIV f x :> D" 
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shows "DERIV (\<lambda>x. f x ^ n) x :> of_nat n * (D * f x ^ (n  Suc 0))" 
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by (cases "n", simp, simp add: DERIV_power_Suc f del: power_Suc) 
21164  210 

29975  211 
text {* Caratheodory formulation of derivative at a point *} 
21164  212 

213 
lemma CARAT_DERIV: 

214 
"(DERIV f x :> l) = 

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(\<exists>g. (\<forall>z. f z  f x = g z * (zx)) & isCont g x & g x = l)" 
21164  216 
(is "?lhs = ?rhs") 
217 
proof 

218 
assume der: "DERIV f x :> l" 

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show "\<exists>g. (\<forall>z. f z  f x = g z * (zx)) \<and> isCont g x \<and> g x = l" 
21164  220 
proof (intro exI conjI) 
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let ?g = "(%z. if z = x then l else (f z  f x) / (zx))" 
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show "\<forall>z. f z  f x = ?g z * (zx)" by simp 
21164  223 
show "isCont ?g x" using der 
224 
by (simp add: isCont_iff DERIV_iff diff_minus 

225 
cong: LIM_equal [rule_format]) 

226 
show "?g x = l" by simp 

227 
qed 

228 
next 

229 
assume "?rhs" 

230 
then obtain g where 

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"(\<forall>z. f z  f x = g z * (zx))" and "isCont g x" and "g x = l" by blast 
21164  232 
thus "(DERIV f x :> l)" 
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by (auto simp add: isCont_iff DERIV_iff cong: LIM_cong) 
21164  234 
qed 
235 

236 
lemma DERIV_chain': 

237 
assumes f: "DERIV f x :> D" 

238 
assumes g: "DERIV g (f x) :> E" 

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shows "DERIV (\<lambda>x. g (f x)) x :> E * D" 
21164  240 
proof (unfold DERIV_iff2) 
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obtain d where d: "\<forall>y. g y  g (f x) = d y * (y  f x)" 
21164  242 
and cont_d: "isCont d (f x)" and dfx: "d (f x) = E" 
243 
using CARAT_DERIV [THEN iffD1, OF g] by fast 

244 
from f have "f  x > f x" 

245 
by (rule DERIV_isCont [unfolded isCont_def]) 

246 
with cont_d have "(\<lambda>z. d (f z))  x > d (f x)" 

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by (rule isCont_tendsto_compose) 
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hence "(\<lambda>z. d (f z) * ((f z  f x) / (z  x))) 
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 x > d (f x) * D" 
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by (rule tendsto_mult [OF _ f [unfolded DERIV_iff2]]) 
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251 
thus "(\<lambda>z. (g (f z)  g (f x)) / (z  x))  x > E * D" 
35216  252 
by (simp add: d dfx) 
21164  253 
qed 
254 

31899  255 
text {* 
256 
Let's do the standard proof, though theorem 

257 
@{text "LIM_mult2"} follows from a NS proof 

258 
*} 

21164  259 

260 
lemma DERIV_cmult: 

261 
"DERIV f x :> D ==> DERIV (%x. c * f x) x :> c*D" 

262 
by (drule DERIV_mult' [OF DERIV_const], simp) 

263 

33654
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264 
lemma DERIV_cdivide: "DERIV f x :> D ==> DERIV (%x. f x / c) x :> D / c" 
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265 
apply (subgoal_tac "DERIV (%x. (1 / c) * f x) x :> (1 / c) * D", force) 
abf780db30ea
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266 
apply (erule DERIV_cmult) 
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267 
done 
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268 

31899  269 
text {* Standard version *} 
21164  270 
lemma DERIV_chain: "[ DERIV f (g x) :> Da; DERIV g x :> Db ] ==> DERIV (f o g) x :> Da * Db" 
35216  271 
by (drule (1) DERIV_chain', simp add: o_def mult_commute) 
21164  272 

273 
lemma DERIV_chain2: "[ DERIV f (g x) :> Da; DERIV g x :> Db ] ==> DERIV (%x. f (g x)) x :> Da * Db" 

274 
by (auto dest: DERIV_chain simp add: o_def) 

275 

31899  276 
text {* Derivative of linear multiplication *} 
21164  277 
lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x :> c" 
23069
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278 
by (cut_tac c = c and x = x in DERIV_ident [THEN DERIV_cmult], simp) 
21164  279 

280 
lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n  Suc 0))" 

23069
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281 
apply (cut_tac DERIV_power [OF DERIV_ident]) 
35216  282 
apply (simp add: real_of_nat_def) 
21164  283 
done 
284 

31899  285 
text {* Power of @{text "1"} *} 
21164  286 

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287 
lemma DERIV_inverse: 
31017  288 
fixes x :: "'a::{real_normed_field}" 
21784
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289 
shows "x \<noteq> 0 ==> DERIV (%x. inverse(x)) x :> ((inverse x ^ Suc (Suc 0)))" 
30273
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290 
by (drule DERIV_inverse' [OF DERIV_ident]) simp 
21164  291 

31899  292 
text {* Derivative of inverse *} 
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293 
lemma DERIV_inverse_fun: 
31017  294 
fixes x :: "'a::{real_normed_field}" 
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295 
shows "[ DERIV f x :> d; f(x) \<noteq> 0 ] 
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296 
==> DERIV (%x. inverse(f x)) x :> ( (d * inverse(f(x) ^ Suc (Suc 0))))" 
30273
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297 
by (drule (1) DERIV_inverse') (simp add: mult_ac nonzero_inverse_mult_distrib) 
21164  298 

31899  299 
text {* Derivative of quotient *} 
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300 
lemma DERIV_quotient: 
31017  301 
fixes x :: "'a::{real_normed_field}" 
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302 
shows "[ DERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 ] 
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303 
==> DERIV (%y. f(y) / (g y)) x :> (d*g(x)  (e*f(x))) / (g(x) ^ Suc (Suc 0))" 
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304 
by (drule (2) DERIV_divide) (simp add: mult_commute) 
21164  305 

31899  306 
text {* @{text "DERIV_intros"} *} 
307 
ML {* 

31902  308 
structure Deriv_Intros = Named_Thms 
31899  309 
( 
45294  310 
val name = @{binding DERIV_intros} 
31899  311 
val description = "DERIV introduction rules" 
312 
) 

313 
*} 

31880  314 

31902  315 
setup Deriv_Intros.setup 
31880  316 

317 
lemma DERIV_cong: "\<lbrakk> DERIV f x :> X ; X = Y \<rbrakk> \<Longrightarrow> DERIV f x :> Y" 

318 
by simp 

319 

320 
declare 

321 
DERIV_const[THEN DERIV_cong, DERIV_intros] 

322 
DERIV_ident[THEN DERIV_cong, DERIV_intros] 

323 
DERIV_add[THEN DERIV_cong, DERIV_intros] 

324 
DERIV_minus[THEN DERIV_cong, DERIV_intros] 

325 
DERIV_mult[THEN DERIV_cong, DERIV_intros] 

326 
DERIV_diff[THEN DERIV_cong, DERIV_intros] 

327 
DERIV_inverse'[THEN DERIV_cong, DERIV_intros] 

328 
DERIV_divide[THEN DERIV_cong, DERIV_intros] 

329 
DERIV_power[where 'a=real, THEN DERIV_cong, 

330 
unfolded real_of_nat_def[symmetric], DERIV_intros] 

331 
DERIV_setsum[THEN DERIV_cong, DERIV_intros] 

22984  332 

31899  333 

22984  334 
subsection {* Differentiability predicate *} 
21164  335 

29169  336 
definition 
337 
differentiable :: "['a::real_normed_field \<Rightarrow> 'a, 'a] \<Rightarrow> bool" 

338 
(infixl "differentiable" 60) where 

339 
"f differentiable x = (\<exists>D. DERIV f x :> D)" 

340 

341 
lemma differentiableE [elim?]: 

342 
assumes "f differentiable x" 

343 
obtains df where "DERIV f x :> df" 

41550  344 
using assms unfolding differentiable_def .. 
29169  345 

21164  346 
lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D" 
347 
by (simp add: differentiable_def) 

348 

349 
lemma differentiableI: "DERIV f x :> D ==> f differentiable x" 

350 
by (force simp add: differentiable_def) 

351 

29169  352 
lemma differentiable_ident [simp]: "(\<lambda>x. x) differentiable x" 
353 
by (rule DERIV_ident [THEN differentiableI]) 

354 

355 
lemma differentiable_const [simp]: "(\<lambda>z. a) differentiable x" 

356 
by (rule DERIV_const [THEN differentiableI]) 

21164  357 

29169  358 
lemma differentiable_compose: 
359 
assumes f: "f differentiable (g x)" 

360 
assumes g: "g differentiable x" 

361 
shows "(\<lambda>x. f (g x)) differentiable x" 

362 
proof  

363 
from `f differentiable (g x)` obtain df where "DERIV f (g x) :> df" .. 

364 
moreover 

365 
from `g differentiable x` obtain dg where "DERIV g x :> dg" .. 

366 
ultimately 

367 
have "DERIV (\<lambda>x. f (g x)) x :> df * dg" by (rule DERIV_chain2) 

368 
thus ?thesis by (rule differentiableI) 

369 
qed 

370 

371 
lemma differentiable_sum [simp]: 

21164  372 
assumes "f differentiable x" 
373 
and "g differentiable x" 

374 
shows "(\<lambda>x. f x + g x) differentiable x" 

375 
proof  

29169  376 
from `f differentiable x` obtain df where "DERIV f x :> df" .. 
377 
moreover 

378 
from `g differentiable x` obtain dg where "DERIV g x :> dg" .. 

379 
ultimately 

380 
have "DERIV (\<lambda>x. f x + g x) x :> df + dg" by (rule DERIV_add) 

381 
thus ?thesis by (rule differentiableI) 

382 
qed 

383 

384 
lemma differentiable_minus [simp]: 

385 
assumes "f differentiable x" 

386 
shows "(\<lambda>x.  f x) differentiable x" 

387 
proof  

388 
from `f differentiable x` obtain df where "DERIV f x :> df" .. 

389 
hence "DERIV (\<lambda>x.  f x) x :>  df" by (rule DERIV_minus) 

390 
thus ?thesis by (rule differentiableI) 

21164  391 
qed 
392 

29169  393 
lemma differentiable_diff [simp]: 
21164  394 
assumes "f differentiable x" 
29169  395 
assumes "g differentiable x" 
21164  396 
shows "(\<lambda>x. f x  g x) differentiable x" 
41550  397 
unfolding diff_minus using assms by simp 
29169  398 

399 
lemma differentiable_mult [simp]: 

400 
assumes "f differentiable x" 

401 
assumes "g differentiable x" 

402 
shows "(\<lambda>x. f x * g x) differentiable x" 

21164  403 
proof  
29169  404 
from `f differentiable x` obtain df where "DERIV f x :> df" .. 
21164  405 
moreover 
29169  406 
from `g differentiable x` obtain dg where "DERIV g x :> dg" .. 
407 
ultimately 

408 
have "DERIV (\<lambda>x. f x * g x) x :> df * g x + dg * f x" by (rule DERIV_mult) 

409 
thus ?thesis by (rule differentiableI) 

21164  410 
qed 
411 

29169  412 
lemma differentiable_inverse [simp]: 
413 
assumes "f differentiable x" and "f x \<noteq> 0" 

414 
shows "(\<lambda>x. inverse (f x)) differentiable x" 

21164  415 
proof  
29169  416 
from `f differentiable x` obtain df where "DERIV f x :> df" .. 
417 
hence "DERIV (\<lambda>x. inverse (f x)) x :>  (inverse (f x) * df * inverse (f x))" 

418 
using `f x \<noteq> 0` by (rule DERIV_inverse') 

419 
thus ?thesis by (rule differentiableI) 

21164  420 
qed 
421 

29169  422 
lemma differentiable_divide [simp]: 
423 
assumes "f differentiable x" 

424 
assumes "g differentiable x" and "g x \<noteq> 0" 

425 
shows "(\<lambda>x. f x / g x) differentiable x" 

41550  426 
unfolding divide_inverse using assms by simp 
29169  427 

428 
lemma differentiable_power [simp]: 

31017  429 
fixes f :: "'a::{real_normed_field} \<Rightarrow> 'a" 
29169  430 
assumes "f differentiable x" 
431 
shows "(\<lambda>x. f x ^ n) differentiable x" 

41550  432 
apply (induct n) 
433 
apply simp 

434 
apply (simp add: assms) 

435 
done 

29169  436 

22984  437 

21164  438 
subsection {* Nested Intervals and Bisection *} 
439 

440 
text{*Lemmas about nested intervals and proof by bisection (cf.Harrison). 

441 
All considerably tidied by lcp.*} 

442 

443 
lemma lemma_f_mono_add [rule_format (no_asm)]: "(\<forall>n. (f::nat=>real) n \<le> f (Suc n)) > f m \<le> f(m + no)" 

444 
apply (induct "no") 

445 
apply (auto intro: order_trans) 

446 
done 

447 

448 
lemma f_inc_g_dec_Beq_f: "[ \<forall>n. f(n) \<le> f(Suc n); 

449 
\<forall>n. g(Suc n) \<le> g(n); 

450 
\<forall>n. f(n) \<le> g(n) ] 

451 
==> Bseq (f :: nat \<Rightarrow> real)" 

452 
apply (rule_tac k = "f 0" and K = "g 0" in BseqI2, rule allI) 

44921  453 
apply (rule conjI) 
21164  454 
apply (induct_tac "n") 
455 
apply (auto intro: order_trans) 

44921  456 
apply (rule_tac y = "g n" in order_trans) 
457 
apply (induct_tac [2] "n") 

21164  458 
apply (auto intro: order_trans) 
459 
done 

460 

461 
lemma f_inc_g_dec_Beq_g: "[ \<forall>n. f(n) \<le> f(Suc n); 

462 
\<forall>n. g(Suc n) \<le> g(n); 

463 
\<forall>n. f(n) \<le> g(n) ] 

464 
==> Bseq (g :: nat \<Rightarrow> real)" 

465 
apply (subst Bseq_minus_iff [symmetric]) 

466 
apply (rule_tac g = "%x.  (f x)" in f_inc_g_dec_Beq_f) 

467 
apply auto 

468 
done 

469 

470 
lemma f_inc_imp_le_lim: 

471 
fixes f :: "nat \<Rightarrow> real" 

472 
shows "\<lbrakk>\<forall>n. f n \<le> f (Suc n); convergent f\<rbrakk> \<Longrightarrow> f n \<le> lim f" 

44921  473 
by (rule incseq_le, simp add: incseq_SucI, simp add: convergent_LIMSEQ_iff) 
21164  474 

31404  475 
lemma lim_uminus: 
476 
fixes g :: "nat \<Rightarrow> 'a::real_normed_vector" 

477 
shows "convergent g ==> lim (%x.  g x) =  (lim g)" 

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478 
apply (rule tendsto_minus [THEN limI]) 
21164  479 
apply (simp add: convergent_LIMSEQ_iff) 
480 
done 

481 

482 
lemma g_dec_imp_lim_le: 

483 
fixes g :: "nat \<Rightarrow> real" 

484 
shows "\<lbrakk>\<forall>n. g (Suc n) \<le> g(n); convergent g\<rbrakk> \<Longrightarrow> lim g \<le> g n" 

44921  485 
by (rule decseq_le, simp add: decseq_SucI, simp add: convergent_LIMSEQ_iff) 
21164  486 

487 
lemma lemma_nest: "[ \<forall>n. f(n) \<le> f(Suc n); 

488 
\<forall>n. g(Suc n) \<le> g(n); 

489 
\<forall>n. f(n) \<le> g(n) ] 

490 
==> \<exists>l m :: real. l \<le> m & ((\<forall>n. f(n) \<le> l) & f > l) & 

491 
((\<forall>n. m \<le> g(n)) & g > m)" 

492 
apply (subgoal_tac "monoseq f & monoseq g") 

493 
prefer 2 apply (force simp add: LIMSEQ_iff monoseq_Suc) 

494 
apply (subgoal_tac "Bseq f & Bseq g") 

495 
prefer 2 apply (blast intro: f_inc_g_dec_Beq_f f_inc_g_dec_Beq_g) 

496 
apply (auto dest!: Bseq_monoseq_convergent simp add: convergent_LIMSEQ_iff) 

497 
apply (rule_tac x = "lim f" in exI) 

498 
apply (rule_tac x = "lim g" in exI) 

499 
apply (auto intro: LIMSEQ_le) 

500 
apply (auto simp add: f_inc_imp_le_lim g_dec_imp_lim_le convergent_LIMSEQ_iff) 

501 
done 

502 

503 
lemma lemma_nest_unique: "[ \<forall>n. f(n) \<le> f(Suc n); 

504 
\<forall>n. g(Suc n) \<le> g(n); 

505 
\<forall>n. f(n) \<le> g(n); 

506 
(%n. f(n)  g(n)) > 0 ] 

507 
==> \<exists>l::real. ((\<forall>n. f(n) \<le> l) & f > l) & 

508 
((\<forall>n. l \<le> g(n)) & g > l)" 

509 
apply (drule lemma_nest, auto) 

510 
apply (subgoal_tac "l = m") 

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discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
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diff
changeset

511 
apply (drule_tac [2] f = f in tendsto_diff) 
21164  512 
apply (auto intro: LIMSEQ_unique) 
513 
done 

514 

515 
text{*The universal quantifiers below are required for the declaration 

516 
of @{text Bolzano_nest_unique} below.*} 

517 

518 
lemma Bolzano_bisect_le: 

519 
"a \<le> b ==> \<forall>n. fst (Bolzano_bisect P a b n) \<le> snd (Bolzano_bisect P a b n)" 

520 
apply (rule allI) 

521 
apply (induct_tac "n") 

522 
apply (auto simp add: Let_def split_def) 

523 
done 

524 

525 
lemma Bolzano_bisect_fst_le_Suc: "a \<le> b ==> 

526 
\<forall>n. fst(Bolzano_bisect P a b n) \<le> fst (Bolzano_bisect P a b (Suc n))" 

527 
apply (rule allI) 

528 
apply (induct_tac "n") 

529 
apply (auto simp add: Bolzano_bisect_le Let_def split_def) 

530 
done 

531 

532 
lemma Bolzano_bisect_Suc_le_snd: "a \<le> b ==> 

533 
\<forall>n. snd(Bolzano_bisect P a b (Suc n)) \<le> snd (Bolzano_bisect P a b n)" 

534 
apply (rule allI) 

535 
apply (induct_tac "n") 

536 
apply (auto simp add: Bolzano_bisect_le Let_def split_def) 

537 
done 

538 

539 
lemma eq_divide_2_times_iff: "((x::real) = y / (2 * z)) = (2 * x = y/z)" 

540 
apply (auto) 

541 
apply (drule_tac f = "%u. (1/2) *u" in arg_cong) 

542 
apply (simp) 

543 
done 

544 

545 
lemma Bolzano_bisect_diff: 

546 
"a \<le> b ==> 

547 
snd(Bolzano_bisect P a b n)  fst(Bolzano_bisect P a b n) = 

548 
(ba) / (2 ^ n)" 

549 
apply (induct "n") 

550 
apply (auto simp add: eq_divide_2_times_iff add_divide_distrib Let_def split_def) 

551 
done 

552 

553 
lemmas Bolzano_nest_unique = 

554 
lemma_nest_unique 

555 
[OF Bolzano_bisect_fst_le_Suc Bolzano_bisect_Suc_le_snd Bolzano_bisect_le] 

556 

557 

558 
lemma not_P_Bolzano_bisect: 

559 
assumes P: "!!a b c. [ P(a,b); P(b,c); a \<le> b; b \<le> c] ==> P(a,c)" 

560 
and notP: "~ P(a,b)" 

561 
and le: "a \<le> b" 

562 
shows "~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))" 

563 
proof (induct n) 

23441  564 
case 0 show ?case using notP by simp 
21164  565 
next 
566 
case (Suc n) 

567 
thus ?case 

568 
by (auto simp del: surjective_pairing [symmetric] 

569 
simp add: Let_def split_def Bolzano_bisect_le [OF le] 

570 
P [of "fst (Bolzano_bisect P a b n)" _ "snd (Bolzano_bisect P a b n)"]) 

571 
qed 

572 

573 
(*Now we repackage P_prem as a formula*) 

574 
lemma not_P_Bolzano_bisect': 

575 
"[ \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c > P(a,c); 

576 
~ P(a,b); a \<le> b ] ==> 

577 
\<forall>n. ~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))" 

578 
by (blast elim!: not_P_Bolzano_bisect [THEN [2] rev_notE]) 

579 

580 

581 

582 
lemma lemma_BOLZANO: 

583 
"[ \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c > P(a,c); 

584 
\<forall>x. \<exists>d::real. 0 < d & 

585 
(\<forall>a b. a \<le> x & x \<le> b & (ba) < d > P(a,b)); 

586 
a \<le> b ] 

587 
==> P(a,b)" 

45600
1bbbac9a0cb0
'lemmas' / 'theorems' commands allow 'for' fixes and standardize the result before storing;
wenzelm
parents:
45294
diff
changeset

588 
apply (rule Bolzano_nest_unique [where P=P, THEN exE], assumption+) 
44568
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discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
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44317
diff
changeset

589 
apply (rule tendsto_minus_cancel) 
21164  590 
apply (simp (no_asm_simp) add: Bolzano_bisect_diff LIMSEQ_divide_realpow_zero) 
591 
apply (rule ccontr) 

592 
apply (drule not_P_Bolzano_bisect', assumption+) 

593 
apply (rename_tac "l") 

594 
apply (drule_tac x = l in spec, clarify) 

31336
e17f13cd1280
generalize constants in SEQ.thy to class metric_space
huffman
parents:
31017
diff
changeset

595 
apply (simp add: LIMSEQ_iff) 
21164  596 
apply (drule_tac P = "%r. 0<r > ?Q r" and x = "d/2" in spec) 
597 
apply (drule_tac P = "%r. 0<r > ?Q r" and x = "d/2" in spec) 

598 
apply (drule real_less_half_sum, auto) 

599 
apply (drule_tac x = "fst (Bolzano_bisect P a b (no + noa))" in spec) 

600 
apply (drule_tac x = "snd (Bolzano_bisect P a b (no + noa))" in spec) 

601 
apply safe 

602 
apply (simp_all (no_asm_simp)) 

603 
apply (rule_tac y = "abs (fst (Bolzano_bisect P a b (no + noa))  l) + abs (snd (Bolzano_bisect P a b (no + noa))  l)" in order_le_less_trans) 

604 
apply (simp (no_asm_simp) add: abs_if) 

605 
apply (rule real_sum_of_halves [THEN subst]) 

606 
apply (rule add_strict_mono) 

607 
apply (simp_all add: diff_minus [symmetric]) 

608 
done 

609 

610 

611 
lemma lemma_BOLZANO2: "((\<forall>a b c. (a \<le> b & b \<le> c & P(a,b) & P(b,c)) > P(a,c)) & 

612 
(\<forall>x. \<exists>d::real. 0 < d & 

613 
(\<forall>a b. a \<le> x & x \<le> b & (ba) < d > P(a,b)))) 

614 
> (\<forall>a b. a \<le> b > P(a,b))" 

615 
apply clarify 

616 
apply (blast intro: lemma_BOLZANO) 

617 
done 

618 

619 

620 
subsection {* Intermediate Value Theorem *} 

621 

622 
text {*Prove Contrapositive by Bisection*} 

623 

624 
lemma IVT: "[ f(a::real) \<le> (y::real); y \<le> f(b); 

625 
a \<le> b; 

626 
(\<forall>x. a \<le> x & x \<le> b > isCont f x) ] 

627 
==> \<exists>x. a \<le> x & x \<le> b & f(x) = y" 

628 
apply (rule contrapos_pp, assumption) 

629 
apply (cut_tac P = "% (u,v) . a \<le> u & u \<le> v & v \<le> b > ~ (f (u) \<le> y & y \<le> f (v))" in lemma_BOLZANO2) 

630 
apply safe 

631 
apply simp_all 

31338
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

632 
apply (simp add: isCont_iff LIM_eq) 
21164  633 
apply (rule ccontr) 
634 
apply (subgoal_tac "a \<le> x & x \<le> b") 

635 
prefer 2 

636 
apply simp 

637 
apply (drule_tac P = "%d. 0<d > ?P d" and x = 1 in spec, arith) 

638 
apply (drule_tac x = x in spec)+ 

639 
apply simp 

640 
apply (drule_tac P = "%r. ?P r > (\<exists>s>0. ?Q r s) " and x = "\<bar>y  f x\<bar>" in spec) 

641 
apply safe 

642 
apply simp 

643 
apply (drule_tac x = s in spec, clarify) 

644 
apply (cut_tac x = "f x" and y = y in linorder_less_linear, safe) 

645 
apply (drule_tac x = "bax" in spec) 

646 
apply (simp_all add: abs_if) 

647 
apply (drule_tac x = "aax" in spec) 

648 
apply (case_tac "x \<le> aa", simp_all) 

649 
done 

650 

651 
lemma IVT2: "[ f(b::real) \<le> (y::real); y \<le> f(a); 

652 
a \<le> b; 

653 
(\<forall>x. a \<le> x & x \<le> b > isCont f x) 

654 
] ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y" 

655 
apply (subgoal_tac " f a \<le> y & y \<le>  f b", clarify) 

656 
apply (drule IVT [where f = "%x.  f x"], assumption) 

44233  657 
apply simp_all 
21164  658 
done 
659 

660 
(*HOL style here: objectlevel formulations*) 

661 
lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b & 

662 
(\<forall>x. a \<le> x & x \<le> b > isCont f x)) 

663 
> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)" 

664 
apply (blast intro: IVT) 

665 
done 

666 

667 
lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b & 

668 
(\<forall>x. a \<le> x & x \<le> b > isCont f x)) 

669 
> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)" 

670 
apply (blast intro: IVT2) 

671 
done 

672 

29975  673 

674 
subsection {* Boundedness of continuous functions *} 

675 

21164  676 
text{*By bisection, function continuous on closed interval is bounded above*} 
677 

678 
lemma isCont_bounded: 

679 
"[ a \<le> b; \<forall>x. a \<le> x & x \<le> b > isCont f x ] 

680 
==> \<exists>M::real. \<forall>x::real. a \<le> x & x \<le> b > f(x) \<le> M" 

681 
apply (cut_tac P = "% (u,v) . a \<le> u & u \<le> v & v \<le> b > (\<exists>M. \<forall>x. u \<le> x & x \<le> v > f x \<le> M)" in lemma_BOLZANO2) 

682 
apply safe 

683 
apply simp_all 

684 
apply (rename_tac x xa ya M Ma) 

36777
be5461582d0f
avoid using realspecific versions of generic lemmas
huffman
parents:
35216
diff
changeset

685 
apply (metis linorder_not_less order_le_less order_trans) 
21164  686 
apply (case_tac "a \<le> x & x \<le> b") 
33654
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

687 
prefer 2 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

688 
apply (rule_tac x = 1 in exI, force) 
31338
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

689 
apply (simp add: LIM_eq isCont_iff) 
21164  690 
apply (drule_tac x = x in spec, auto) 
691 
apply (erule_tac V = "\<forall>M. \<exists>x. a \<le> x & x \<le> b & ~ f x \<le> M" in thin_rl) 

692 
apply (drule_tac x = 1 in spec, auto) 

693 
apply (rule_tac x = s in exI, clarify) 

694 
apply (rule_tac x = "\<bar>f x\<bar> + 1" in exI, clarify) 

695 
apply (drule_tac x = "xax" in spec) 

696 
apply (auto simp add: abs_ge_self) 

697 
done 

698 

699 
text{*Refine the above to existence of least upper bound*} 

700 

701 
lemma lemma_reals_complete: "((\<exists>x. x \<in> S) & (\<exists>y. isUb UNIV S (y::real))) > 

702 
(\<exists>t. isLub UNIV S t)" 

703 
by (blast intro: reals_complete) 

704 

705 
lemma isCont_has_Ub: "[ a \<le> b; \<forall>x. a \<le> x & x \<le> b > isCont f x ] 

706 
==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b > f(x) \<le> M) & 

707 
(\<forall>N. N < M > (\<exists>x. a \<le> x & x \<le> b & N < f(x)))" 

708 
apply (cut_tac S = "Collect (%y. \<exists>x. a \<le> x & x \<le> b & y = f x)" 

709 
in lemma_reals_complete) 

710 
apply auto 

711 
apply (drule isCont_bounded, assumption) 

712 
apply (auto simp add: isUb_def leastP_def isLub_def setge_def setle_def) 

713 
apply (rule exI, auto) 

714 
apply (auto dest!: spec simp add: linorder_not_less) 

715 
done 

716 

717 
text{*Now show that it attains its upper bound*} 

718 

719 
lemma isCont_eq_Ub: 

720 
assumes le: "a \<le> b" 

721 
and con: "\<forall>x::real. a \<le> x & x \<le> b > isCont f x" 

722 
shows "\<exists>M::real. (\<forall>x. a \<le> x & x \<le> b > f(x) \<le> M) & 

723 
(\<exists>x. a \<le> x & x \<le> b & f(x) = M)" 

724 
proof  

725 
from isCont_has_Ub [OF le con] 

726 
obtain M where M1: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M" 

727 
and M2: "!!N. N<M ==> \<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x" by blast 

728 
show ?thesis 

729 
proof (intro exI, intro conjI) 

730 
show " \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M" by (rule M1) 

731 
show "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M" 

732 
proof (rule ccontr) 

733 
assume "\<not> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)" 

734 
with M1 have M3: "\<forall>x. a \<le> x & x \<le> b > f x < M" 

44890
22f665a2e91c
new fastforce replacing fastsimp  less confusing name
nipkow
parents:
44568
diff
changeset

735 
by (fastforce simp add: linorder_not_le [symmetric]) 
21164  736 
hence "\<forall>x. a \<le> x & x \<le> b > isCont (%x. inverse (M  f x)) x" 
44233  737 
by (auto simp add: con) 
21164  738 
from isCont_bounded [OF le this] 
739 
obtain k where k: "!!x. a \<le> x & x \<le> b > inverse (M  f x) \<le> k" by auto 

740 
have Minv: "!!x. a \<le> x & x \<le> b > 0 < inverse (M  f (x))" 

29667  741 
by (simp add: M3 algebra_simps) 
21164  742 
have "!!x. a \<le> x & x \<le> b > inverse (M  f x) < k+1" using k 
743 
by (auto intro: order_le_less_trans [of _ k]) 

744 
with Minv 

745 
have "!!x. a \<le> x & x \<le> b > inverse(k+1) < inverse(inverse(M  f x))" 

746 
by (intro strip less_imp_inverse_less, simp_all) 

747 
hence invlt: "!!x. a \<le> x & x \<le> b > inverse(k+1) < M  f x" 

748 
by simp 

749 
have "M  inverse (k+1) < M" using k [of a] Minv [of a] le 

750 
by (simp, arith) 

751 
from M2 [OF this] 

752 
obtain x where ax: "a \<le> x & x \<le> b & M  inverse(k+1) < f x" .. 

753 
thus False using invlt [of x] by force 

754 
qed 

755 
qed 

756 
qed 

757 

758 

759 
text{*Same theorem for lower bound*} 

760 

761 
lemma isCont_eq_Lb: "[ a \<le> b; \<forall>x. a \<le> x & x \<le> b > isCont f x ] 

762 
==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b > M \<le> f(x)) & 

763 
(\<exists>x. a \<le> x & x \<le> b & f(x) = M)" 

764 
apply (subgoal_tac "\<forall>x. a \<le> x & x \<le> b > isCont (%x.  (f x)) x") 

765 
prefer 2 apply (blast intro: isCont_minus) 

766 
apply (drule_tac f = "(%x.  (f x))" in isCont_eq_Ub) 

767 
apply safe 

768 
apply auto 

769 
done 

770 

771 

772 
text{*Another version.*} 

773 

774 
lemma isCont_Lb_Ub: "[a \<le> b; \<forall>x. a \<le> x & x \<le> b > isCont f x ] 

775 
==> \<exists>L M::real. (\<forall>x::real. a \<le> x & x \<le> b > L \<le> f(x) & f(x) \<le> M) & 

776 
(\<forall>y. L \<le> y & y \<le> M > (\<exists>x. a \<le> x & x \<le> b & (f(x) = y)))" 

777 
apply (frule isCont_eq_Lb) 

778 
apply (frule_tac [2] isCont_eq_Ub) 

779 
apply (assumption+, safe) 

780 
apply (rule_tac x = "f x" in exI) 

781 
apply (rule_tac x = "f xa" in exI, simp, safe) 

782 
apply (cut_tac x = x and y = xa in linorder_linear, safe) 

783 
apply (cut_tac f = f and a = x and b = xa and y = y in IVT_objl) 

784 
apply (cut_tac [2] f = f and a = xa and b = x and y = y in IVT2_objl, safe) 

785 
apply (rule_tac [2] x = xb in exI) 

786 
apply (rule_tac [4] x = xb in exI, simp_all) 

787 
done 

788 

789 

29975  790 
subsection {* Local extrema *} 
791 

21164  792 
text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*} 
793 

33654
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

794 
lemma DERIV_pos_inc_right: 
21164  795 
fixes f :: "real => real" 
796 
assumes der: "DERIV f x :> l" 

797 
and l: "0 < l" 

798 
shows "\<exists>d > 0. \<forall>h > 0. h < d > f(x) < f(x + h)" 

799 
proof  

800 
from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]] 

801 
have "\<exists>s > 0. (\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z)  f x) / z  l\<bar> < l)" 

802 
by (simp add: diff_minus) 

803 
then obtain s 

804 
where s: "0 < s" 

805 
and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z)  f x) / z  l\<bar> < l" 

806 
by auto 

807 
thus ?thesis 

808 
proof (intro exI conjI strip) 

23441  809 
show "0<s" using s . 
21164  810 
fix h::real 
811 
assume "0 < h" "h < s" 

812 
with all [of h] show "f x < f (x+h)" 

813 
proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric] 

814 
split add: split_if_asm) 

815 
assume "~ (f (x+h)  f x) / h < l" and h: "0 < h" 

816 
with l 

817 
have "0 < (f (x+h)  f x) / h" by arith 

818 
thus "f x < f (x+h)" 

819 
by (simp add: pos_less_divide_eq h) 

820 
qed 

821 
qed 

822 
qed 

823 

33654
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

824 
lemma DERIV_neg_dec_left: 
21164  825 
fixes f :: "real => real" 
826 
assumes der: "DERIV f x :> l" 

827 
and l: "l < 0" 

828 
shows "\<exists>d > 0. \<forall>h > 0. h < d > f(x) < f(xh)" 

829 
proof  

830 
from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]] 

831 
have "\<exists>s > 0. (\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z)  f x) / z  l\<bar> < l)" 

832 
by (simp add: diff_minus) 

833 
then obtain s 

834 
where s: "0 < s" 

835 
and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z)  f x) / z  l\<bar> < l" 

836 
by auto 

837 
thus ?thesis 

838 
proof (intro exI conjI strip) 

23441  839 
show "0<s" using s . 
21164  840 
fix h::real 
841 
assume "0 < h" "h < s" 

842 
with all [of "h"] show "f x < f (xh)" 

843 
proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric] 

844 
split add: split_if_asm) 

845 
assume "  ((f (xh)  f x) / h) < l" and h: "0 < h" 

846 
with l 

847 
have "0 < (f (xh)  f x) / h" by arith 

848 
thus "f x < f (xh)" 

849 
by (simp add: pos_less_divide_eq h) 

850 
qed 

851 
qed 

852 
qed 

853 

33654
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

854 

abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

855 
lemma DERIV_pos_inc_left: 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

856 
fixes f :: "real => real" 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

857 
shows "DERIV f x :> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d > f(x  h) < f(x)" 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

858 
apply (rule DERIV_neg_dec_left [of "%x.  f x" x "l", simplified]) 
41368  859 
apply (auto simp add: DERIV_minus) 
33654
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

860 
done 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

861 

abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

862 
lemma DERIV_neg_dec_right: 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

863 
fixes f :: "real => real" 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

864 
shows "DERIV f x :> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d > f(x) > f(x + h)" 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

865 
apply (rule DERIV_pos_inc_right [of "%x.  f x" x "l", simplified]) 
41368  866 
apply (auto simp add: DERIV_minus) 
33654
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

867 
done 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

868 

21164  869 
lemma DERIV_local_max: 
870 
fixes f :: "real => real" 

871 
assumes der: "DERIV f x :> l" 

872 
and d: "0 < d" 

873 
and le: "\<forall>y. \<bar>xy\<bar> < d > f(y) \<le> f(x)" 

874 
shows "l = 0" 

875 
proof (cases rule: linorder_cases [of l 0]) 

23441  876 
case equal thus ?thesis . 
21164  877 
next 
878 
case less 

33654
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

879 
from DERIV_neg_dec_left [OF der less] 
21164  880 
obtain d' where d': "0 < d'" 
881 
and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (xh)" by blast 

882 
from real_lbound_gt_zero [OF d d'] 

883 
obtain e where "0 < e \<and> e < d \<and> e < d'" .. 

884 
with lt le [THEN spec [where x="xe"]] 

885 
show ?thesis by (auto simp add: abs_if) 

886 
next 

887 
case greater 

33654
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

888 
from DERIV_pos_inc_right [OF der greater] 
21164  889 
obtain d' where d': "0 < d'" 
890 
and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast 

891 
from real_lbound_gt_zero [OF d d'] 

892 
obtain e where "0 < e \<and> e < d \<and> e < d'" .. 

893 
with lt le [THEN spec [where x="x+e"]] 

894 
show ?thesis by (auto simp add: abs_if) 

895 
qed 

896 

897 

898 
text{*Similar theorem for a local minimum*} 

899 
lemma DERIV_local_min: 

900 
fixes f :: "real => real" 

901 
shows "[ DERIV f x :> l; 0 < d; \<forall>y. \<bar>xy\<bar> < d > f(x) \<le> f(y) ] ==> l = 0" 

902 
by (drule DERIV_minus [THEN DERIV_local_max], auto) 

903 

904 

905 
text{*In particular, if a function is locally flat*} 

906 
lemma DERIV_local_const: 

907 
fixes f :: "real => real" 

908 
shows "[ DERIV f x :> l; 0 < d; \<forall>y. \<bar>xy\<bar> < d > f(x) = f(y) ] ==> l = 0" 

909 
by (auto dest!: DERIV_local_max) 

910 

29975  911 

912 
subsection {* Rolle's Theorem *} 

913 

21164  914 
text{*Lemma about introducing open ball in open interval*} 
915 
lemma lemma_interval_lt: 

916 
"[ a < x; x < b ] 

917 
==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>xy\<bar> < d > a < y & y < b)" 

27668  918 

22998  919 
apply (simp add: abs_less_iff) 
21164  920 
apply (insert linorder_linear [of "xa" "bx"], safe) 
921 
apply (rule_tac x = "xa" in exI) 

922 
apply (rule_tac [2] x = "bx" in exI, auto) 

923 
done 

924 

925 
lemma lemma_interval: "[ a < x; x < b ] ==> 

926 
\<exists>d::real. 0 < d & (\<forall>y. \<bar>xy\<bar> < d > a \<le> y & y \<le> b)" 

927 
apply (drule lemma_interval_lt, auto) 

44921  928 
apply force 
21164  929 
done 
930 

931 
text{*Rolle's Theorem. 

932 
If @{term f} is defined and continuous on the closed interval 

933 
@{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"}, 

934 
and @{term "f(a) = f(b)"}, 

935 
then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*} 

936 
theorem Rolle: 

937 
assumes lt: "a < b" 

938 
and eq: "f(a) = f(b)" 

939 
and con: "\<forall>x. a \<le> x & x \<le> b > isCont f x" 

940 
and dif [rule_format]: "\<forall>x. a < x & x < b > f differentiable x" 

21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset

941 
shows "\<exists>z::real. a < z & z < b & DERIV f z :> 0" 
21164  942 
proof  
943 
have le: "a \<le> b" using lt by simp 

944 
from isCont_eq_Ub [OF le con] 

945 
obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x" 

946 
and alex: "a \<le> x" and xleb: "x \<le> b" 

947 
by blast 

948 
from isCont_eq_Lb [OF le con] 

949 
obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z" 

950 
and alex': "a \<le> x'" and x'leb: "x' \<le> b" 

951 
by blast 

952 
show ?thesis 

953 
proof cases 

954 
assume axb: "a < x & x < b" 

955 
{*@{term f} attains its maximum within the interval*} 

27668  956 
hence ax: "a<x" and xb: "x<b" by arith + 
21164  957 
from lemma_interval [OF ax xb] 
958 
obtain d where d: "0<d" and bound: "\<forall>y. \<bar>xy\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b" 

959 
by blast 

960 
hence bound': "\<forall>y. \<bar>xy\<bar> < d \<longrightarrow> f y \<le> f x" using x_max 

961 
by blast 

962 
from differentiableD [OF dif [OF axb]] 

963 
obtain l where der: "DERIV f x :> l" .. 

964 
have "l=0" by (rule DERIV_local_max [OF der d bound']) 

965 
{*the derivative at a local maximum is zero*} 

966 
thus ?thesis using ax xb der by auto 

967 
next 

968 
assume notaxb: "~ (a < x & x < b)" 

969 
hence xeqab: "x=a  x=b" using alex xleb by arith 

970 
hence fb_eq_fx: "f b = f x" by (auto simp add: eq) 

971 
show ?thesis 

972 
proof cases 

973 
assume ax'b: "a < x' & x' < b" 

974 
{*@{term f} attains its minimum within the interval*} 

27668  975 
hence ax': "a<x'" and x'b: "x'<b" by arith+ 
21164  976 
from lemma_interval [OF ax' x'b] 
977 
obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b" 

978 
by blast 

979 
hence bound': "\<forall>y. \<bar>x'y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min 

980 
by blast 

981 
from differentiableD [OF dif [OF ax'b]] 

982 
obtain l where der: "DERIV f x' :> l" .. 

983 
have "l=0" by (rule DERIV_local_min [OF der d bound']) 

984 
{*the derivative at a local minimum is zero*} 

985 
thus ?thesis using ax' x'b der by auto 

986 
next 

987 
assume notax'b: "~ (a < x' & x' < b)" 

988 
{*@{term f} is constant througout the interval*} 

989 
hence x'eqab: "x'=a  x'=b" using alex' x'leb by arith 

990 
hence fb_eq_fx': "f b = f x'" by (auto simp add: eq) 

991 
from dense [OF lt] 

992 
obtain r where ar: "a < r" and rb: "r < b" by blast 

993 
from lemma_interval [OF ar rb] 

994 
obtain d where d: "0<d" and bound: "\<forall>y. \<bar>ry\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b" 

995 
by blast 

996 
have eq_fb: "\<forall>z. a \<le> z > z \<le> b > f z = f b" 

997 
proof (clarify) 

998 
fix z::real 

999 
assume az: "a \<le> z" and zb: "z \<le> b" 

1000 
show "f z = f b" 

1001 
proof (rule order_antisym) 

1002 
show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb) 

1003 
show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb) 

1004 
qed 

1005 
qed 

1006 
have bound': "\<forall>y. \<bar>ry\<bar> < d \<longrightarrow> f r = f y" 

1007 
proof (intro strip) 

1008 
fix y::real 

1009 
assume lt: "\<bar>ry\<bar> < d" 

1010 
hence "f y = f b" by (simp add: eq_fb bound) 

1011 
thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le) 

1012 
qed 

1013 
from differentiableD [OF dif [OF conjI [OF ar rb]]] 

1014 
obtain l where der: "DERIV f r :> l" .. 

1015 
have "l=0" by (rule DERIV_local_const [OF der d bound']) 

1016 
{*the derivative of a constant function is zero*} 

1017 
thus ?thesis using ar rb der by auto 

1018 
qed 

1019 
qed 

1020 
qed 

1021 

1022 

1023 
subsection{*Mean Value Theorem*} 

1024 

1025 
lemma lemma_MVT: 

1026 
"f a  (f b  f a)/(ba) * a = f b  (f b  f a)/(ba) * (b::real)" 

1027 
proof cases 

1028 
assume "a=b" thus ?thesis by simp 

1029 
next 

1030 
assume "a\<noteq>b" 

1031 
hence ba: "ba \<noteq> 0" by arith 

1032 
show ?thesis 

1033 
by (rule real_mult_left_cancel [OF ba, THEN iffD1], 

1034 
simp add: right_diff_distrib, 

1035 
simp add: left_diff_distrib) 

1036 
qed 

1037 

1038 
theorem MVT: 

1039 
assumes lt: "a < b" 

1040 
and con: "\<forall>x. a \<le> x & x \<le> b > isCont f x" 

1041 
and dif [rule_format]: "\<forall>x. a < x & x < b > f differentiable x" 

21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset

1042 
shows "\<exists>l z::real. a < z & z < b & DERIV f z :> l & 
21164  1043 
(f(b)  f(a) = (ba) * l)" 
1044 
proof  

1045 
let ?F = "%x. f x  ((f b  f a) / (ba)) * x" 

44233  1046 
have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x" 
1047 
using con by (fast intro: isCont_intros) 

21164  1048 
have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable x" 
1049 
proof (clarify) 

1050 
fix x::real 

1051 
assume ax: "a < x" and xb: "x < b" 

1052 
from differentiableD [OF dif [OF conjI [OF ax xb]]] 

1053 
obtain l where der: "DERIV f x :> l" .. 

1054 
show "?F differentiable x" 

1055 
by (rule differentiableI [where D = "l  (f b  f a)/(ba)"], 

1056 
blast intro: DERIV_diff DERIV_cmult_Id der) 

1057 
qed 

1058 
from Rolle [where f = ?F, OF lt lemma_MVT contF difF] 

1059 
obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0" 

1060 
by blast 

1061 
have "DERIV (%x. ((f b  f a)/(ba)) * x) z :> (f b  f a)/(ba)" 

1062 
by (rule DERIV_cmult_Id) 

1063 
hence derF: "DERIV (\<lambda>x. ?F x + (f b  f a) / (b  a) * x) z 

1064 
:> 0 + (f b  f a) / (b  a)" 

1065 
by (rule DERIV_add [OF der]) 

1066 
show ?thesis 

1067 
proof (intro exI conjI) 

23441  1068 
show "a < z" using az . 
1069 
show "z < b" using zb . 

21164  1070 
show "f b  f a = (b  a) * ((f b  f a)/(ba))" by (simp) 
1071 
show "DERIV f z :> ((f b  f a)/(ba))" using derF by simp 

1072 
qed 

1073 
qed 

1074 

29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1075 
lemma MVT2: 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1076 
"[ a < b; \<forall>x. a \<le> x & x \<le> b > DERIV f x :> f'(x) ] 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1077 
==> \<exists>z::real. a < z & z < b & (f b  f a = (b  a) * f'(z))" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1078 
apply (drule MVT) 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1079 
apply (blast intro: DERIV_isCont) 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1080 
apply (force dest: order_less_imp_le simp add: differentiable_def) 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1081 
apply (blast dest: DERIV_unique order_less_imp_le) 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1082 
done 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1083 

21164  1084 

1085 
text{*A function is constant if its derivative is 0 over an interval.*} 

1086 

1087 
lemma DERIV_isconst_end: 

1088 
fixes f :: "real => real" 

1089 
shows "[ a < b; 

1090 
\<forall>x. a \<le> x & x \<le> b > isCont f x; 

1091 
\<forall>x. a < x & x < b > DERIV f x :> 0 ] 

1092 
==> f b = f a" 

1093 
apply (drule MVT, assumption) 

1094 
apply (blast intro: differentiableI) 

1095 
apply (auto dest!: DERIV_unique simp add: diff_eq_eq) 

1096 
done 

1097 

1098 
lemma DERIV_isconst1: 

1099 
fixes f :: "real => real" 

1100 
shows "[ a < b; 

1101 
\<forall>x. a \<le> x & x \<le> b > isCont f x; 

1102 
\<forall>x. a < x & x < b > DERIV f x :> 0 ] 

1103 
==> \<forall>x. a \<le> x & x \<le> b > f x = f a" 

1104 
apply safe 

1105 
apply (drule_tac x = a in order_le_imp_less_or_eq, safe) 

1106 
apply (drule_tac b = x in DERIV_isconst_end, auto) 

1107 
done 

1108 

1109 
lemma DERIV_isconst2: 

1110 
fixes f :: "real => real" 

1111 
shows "[ a < b; 

1112 
\<forall>x. a \<le> x & x \<le> b > isCont f x; 

1113 
\<forall>x. a < x & x < b > DERIV f x :> 0; 

1114 
a \<le> x; x \<le> b ] 

1115 
==> f x = f a" 

1116 
apply (blast dest: DERIV_isconst1) 

1117 
done 

1118 

29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1119 
lemma DERIV_isconst3: fixes a b x y :: real 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1120 
assumes "a < b" and "x \<in> {a <..< b}" and "y \<in> {a <..< b}" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1121 
assumes derivable: "\<And>x. x \<in> {a <..< b} \<Longrightarrow> DERIV f x :> 0" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1122 
shows "f x = f y" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1123 
proof (cases "x = y") 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1124 
case False 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1125 
let ?a = "min x y" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1126 
let ?b = "max x y" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1127 

c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1128 
have "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> DERIV f z :> 0" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1129 
proof (rule allI, rule impI) 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1130 
fix z :: real assume "?a \<le> z \<and> z \<le> ?b" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1131 
hence "a < z" and "z < b" using `x \<in> {a <..< b}` and `y \<in> {a <..< b}` by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1132 
hence "z \<in> {a<..<b}" by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1133 
thus "DERIV f z :> 0" by (rule derivable) 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1134 
qed 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1135 
hence isCont: "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> isCont f z" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1136 
and DERIV: "\<forall>z. ?a < z \<and> z < ?b \<longrightarrow> DERIV f z :> 0" using DERIV_isCont by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1137 

c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1138 
have "?a < ?b" using `x \<noteq> y` by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1139 
from DERIV_isconst2[OF this isCont DERIV, of x] and DERIV_isconst2[OF this isCont DERIV, of y] 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1140 
show ?thesis by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1141 
qed auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1142 

21164  1143 
lemma DERIV_isconst_all: 
1144 
fixes f :: "real => real" 

1145 
shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)" 

1146 
apply (rule linorder_cases [of x y]) 

1147 
apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+ 

1148 
done 

1149 

1150 
lemma DERIV_const_ratio_const: 

21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset

1151 
fixes f :: "real => real" 
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset

1152 
shows "[a \<noteq> b; \<forall>x. DERIV f x :> k ] ==> (f(b)  f(a)) = (ba) * k" 
21164  1153 
apply (rule linorder_cases [of a b], auto) 
1154 
apply (drule_tac [!] f = f in MVT) 

1155 
apply (auto dest: DERIV_isCont DERIV_unique simp add: differentiable_def) 

23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23441
diff
changeset

1156 
apply (auto dest: DERIV_unique simp add: ring_distribs diff_minus) 
21164  1157 
done 
1158 

1159 
lemma DERIV_const_ratio_const2: 

21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset

1160 
fixes f :: "real => real" 
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset

1161 
shows "[a \<noteq> b; \<forall>x. DERIV f x :> k ] ==> (f(b)  f(a))/(ba) = k" 
21164  1162 
apply (rule_tac c1 = "ba" in real_mult_right_cancel [THEN iffD1]) 
1163 
apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc) 

1164 
done 

1165 

1166 
lemma real_average_minus_first [simp]: "((a + b) /2  a) = (ba)/(2::real)" 

1167 
by (simp) 

1168 

1169 
lemma real_average_minus_second [simp]: "((b + a)/2  a) = (ba)/(2::real)" 

1170 
by (simp) 

1171 

1172 
text{*Gallileo's "trick": average velocity = av. of end velocities*} 

1173 

1174 
lemma DERIV_const_average: 

1175 
fixes v :: "real => real" 

1176 
assumes neq: "a \<noteq> (b::real)" 

1177 
and der: "\<forall>x. DERIV v x :> k" 

1178 
shows "v ((a + b)/2) = (v a + v b)/2" 

1179 
proof (cases rule: linorder_cases [of a b]) 

1180 
case equal with neq show ?thesis by simp 

1181 
next 

1182 
case less 

1183 
have "(v b  v a) / (b  a) = k" 

1184 
by (rule DERIV_const_ratio_const2 [OF neq der]) 

1185 
hence "(ba) * ((v b  v a) / (ba)) = (ba) * k" by simp 

1186 
moreover have "(v ((a + b) / 2)  v a) / ((a + b) / 2  a) = k" 

1187 
by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq) 

1188 
ultimately show ?thesis using neq by force 

1189 
next 

1190 
case greater 

1191 
have "(v b  v a) / (b  a) = k" 

1192 
by (rule DERIV_const_ratio_const2 [OF neq der]) 

1193 
hence "(ba) * ((v b  v a) / (ba)) = (ba) * k" by simp 

1194 
moreover have " (v ((b + a) / 2)  v a) / ((b + a) / 2  a) = k" 

1195 
by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq) 

1196 
ultimately show ?thesis using neq by (force simp add: add_commute) 

1197 
qed 

1198 

33654
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1199 
(* A function with positive derivative is increasing. 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1200 
A simple proof using the MVT, by Jeremy Avigad. And variants. 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1201 
*) 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1202 
lemma DERIV_pos_imp_increasing: 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1203 
fixes a::real and b::real and f::"real => real" 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1204 
assumes "a < b" and "\<forall>x. a \<le> x & x \<le> b > (EX y. DERIV f x :> y & y > 0)" 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1205 
shows "f a < f b" 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1206 
proof (rule ccontr) 
41550  1207 
assume f: "~ f a < f b" 
33690  1208 
have "EX l z. a < z & z < b & DERIV f z :> l 
33654
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1209 
& f b  f a = (b  a) * l" 
33690  1210 
apply (rule MVT) 
1211 
using assms 

1212 
apply auto 

1213 
apply (metis DERIV_isCont) 

36777
be5461582d0f
avoid using realspecific versions of generic lemmas
huffman
parents:
35216
diff
changeset

1214 
apply (metis differentiableI less_le) 
33690  1215 
done 
41550  1216 
then obtain l z where z: "a < z" "z < b" "DERIV f z :> l" 
33654
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1217 
and "f b  f a = (b  a) * l" 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1218 
by auto 
41550  1219 
with assms f have "~(l > 0)" 
36777
be5461582d0f
avoid using realspecific versions of generic lemmas
huffman
parents:
35216
diff
changeset

1220 
by (metis linorder_not_le mult_le_0_iff diff_le_0_iff_le) 
41550  1221 
with assms z show False 
36777
be5461582d0f
avoid using realspecific versions of generic lemmas
huffman
parents:
35216
diff
changeset

1222 
by (metis DERIV_unique less_le) 
33654
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1223 
qed 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1224 

45791  1225 
lemma DERIV_nonneg_imp_nondecreasing: 
33654
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1226 
fixes a::real and b::real and f::"real => real" 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1227 
assumes "a \<le> b" and 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1228 
"\<forall>x. a \<le> x & x \<le> b > (\<exists>y. DERIV f x :> y & y \<ge> 0)" 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1229 
shows "f a \<le> f b" 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1230 
proof (rule ccontr, cases "a = b") 
41550  1231 
assume "~ f a \<le> f b" and "a = b" 
1232 
then show False by auto 

37891  1233 
next 
1234 
assume A: "~ f a \<le> f b" 

1235 
assume B: "a ~= b" 

33654
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1236 
with assms have "EX l z. a < z & z < b & DERIV f z :> l 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1237 
& f b  f a = (b  a) * l" 
33690  1238 
apply  
1239 
apply (rule MVT) 

1240 
apply auto 

1241 
apply (metis DERIV_isCont) 

36777
be5461582d0f
avoid using realspecific versions of generic lemmas
huffman
parents:
35216
diff
changeset

1242 
apply (metis differentiableI less_le) 
33654
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1243 
done 
41550  1244 
then obtain l z where z: "a < z" "z < b" "DERIV f z :> l" 
37891  1245 
and C: "f b  f a = (b  a) * l" 
33654
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1246 
by auto 
37891  1247 
with A have "a < b" "f b < f a" by auto 
1248 
with C have "\<not> l \<ge> 0" by (auto simp add: not_le algebra_simps) 

45051
c478d1876371
discontinued legacy theorem names from RealDef.thy
huffman
parents:
44921
diff
changeset

1249 
(metis A add_le_cancel_right assms(1) less_eq_real_def mult_right_mono add_left_mono linear order_refl) 
41550  1250 
with assms z show False 
33654
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1251 
by (metis DERIV_unique order_less_imp_le) 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1252 
qed 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1253 

abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1254 
lemma DERIV_neg_imp_decreasing: 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1255 
fixes a::real and b::real and f::"real => real" 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1256 
assumes "a < b" and 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1257 
"\<forall>x. a \<le> x & x \<le> b > (\<exists>y. DERIV f x :> y & y < 0)" 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1258 
shows "f a > f b" 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1259 
proof  
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1260 
have "(%x. f x) a < (%x. f x) b" 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1261 
apply (rule DERIV_pos_imp_increasing [of a b "%x. f x"]) 
33690  1262 
using assms 
1263 
apply auto 

33654
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1264 
apply (metis DERIV_minus neg_0_less_iff_less) 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1265 
done 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1266 
thus ?thesis 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1267 
by simp 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1268 
qed 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1269 

abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1270 
lemma DERIV_nonpos_imp_nonincreasing: 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1271 
fixes a::real and b::real and f::"real => real" 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1272 
assumes "a \<le> b" and 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1273 
"\<forall>x. a \<le> x & x \<le> b > (\<exists>y. DERIV f x :> y & y \<le> 0)" 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1274 
shows "f a \<ge> f b" 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1275 
proof  
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1276 
have "(%x. f x) a \<le> (%x. f x) b" 
45791  1277 
apply (rule DERIV_nonneg_imp_nondecreasing [of a b "%x. f x"]) 
33690  1278 
using assms 
1279 
apply auto 

33654
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1280 
apply (metis DERIV_minus neg_0_le_iff_le) 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1281 
done 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1282 
thus ?thesis 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1283 
by simp 
abf780db30ea
A number of theorems contributed by Jeremy Avigad
paulson
parents:
31902
diff
changeset

1284 
qed 
21164  1285 

29975  1286 
subsection {* Continuous injective functions *} 
1287 

21164  1288 
text{*Dull lemma: an continuous injection on an interval must have a 
1289 
strict maximum at an end point, not in the middle.*} 

1290 

1291 
lemma lemma_isCont_inj: 

1292 
fixes f :: "real \<Rightarrow> real" 

1293 
assumes d: "0 < d" 

1294 
and inj [rule_format]: "\<forall>z. \<bar>zx\<bar> \<le> d > g(f z) = z" 

1295 
and cont: "\<forall>z. \<bar>zx\<bar> \<le> d > isCont f z" 

1296 
shows "\<exists>z. \<bar>zx\<bar> \<le> d & f x < f z" 

1297 
proof (rule ccontr) 

1298 
assume "~ (\<exists>z. \<bar>zx\<bar> \<le> d & f x < f z)" 

1299 
hence all [rule_format]: "\<forall>z. \<bar>z  x\<bar> \<le> d > f z \<le> f x" by auto 

1300 
show False 

1301 
proof (cases rule: linorder_le_cases [of "f(xd)" "f(x+d)"]) 

1302 
case le 

1303 
from d cont all [of "x+d"] 

1304 
have flef: "f(x+d) \<le> f x" 

1305 
and xlex: "x  d \<le> x" 

1306 
and cont': "\<forall>z. x  d \<le> z \<and> z \<le> x \<longrightarrow> isCont f z" 

1307 
by (auto simp add: abs_if) 

1308 
from IVT [OF le flef xlex cont'] 

1309 
obtain x' where "xd \<le> x'" "x' \<le> x" "f x' = f(x+d)" by blast 

1310 
moreover 

1311 
hence "g(f x') = g (f(x+d))" by simp 

1312 
ultimately show False using d inj [of x'] inj [of "x+d"] 

22998  1313 
by (simp add: abs_le_iff) 
21164  1314 
next 
1315 
case ge 

1316 
from d cont all [of "xd"] 

1317 
have flef: "f(xd) \<le> f x" 

1318 
and xlex: "x \<le> x+d" 

1319 
and cont': "\<forall>z. x \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z" 

1320 
by (auto simp add: abs_if) 

1321 
from IVT2 [OF ge flef xlex cont'] 

1322 
obtain x' where "x \<le> x'" "x' \<le> x+d" "f x' = f(xd)" by blast 

1323 
moreover 

1324 
hence "g(f x') = g (f(xd))" by simp 

1325 
ultimately show False using d inj [of x'] inj [of "xd"] 

22998  1326 
by (simp add: abs_le_iff) 
21164  1327 
qed 
1328 
qed 

1329 

1330 

1331 
text{*Similar version for lower bound.*} 

1332 

1333 
lemma lemma_isCont_inj2: 

1334 
fixes f g :: "real \<Rightarrow> real" 

1335 
shows "[0 < d; \<forall>z. \<bar>zx\<bar> \<le> d > g(f z) = z; 

1336 
\<forall>z. \<bar>zx\<bar> \<le> d > isCont f z ] 

1337 
==> \<exists>z. \<bar>zx\<bar> \<le> d & f z < f x" 

1338 
apply (insert lemma_isCont_inj 

1339 
[where f = "%x.  f x" and g = "%y. g(y)" and x = x and d = d]) 

44233  1340 
apply (simp add: linorder_not_le) 
21164  1341 
done 
1342 

1343 
text{*Show there's an interval surrounding @{term "f(x)"} in 

1344 
@{text "f[[x  d, x + d]]"} .*} 

1345 

1346 
lemma isCont_inj_range: 

1347 
fixes f :: "real \<Rightarrow> real" 

1348 
assumes d: "0 < d" 

1349 
and inj: "\<forall>z. \<bar>zx\<bar> \<le> d > g(f z) = z" 

1350 
and cont: "\<forall>z. \<bar>zx\<bar> \<le> d > isCont f z" 

1351 
shows "\<exists>e>0. \<forall>y. \<bar>y  f x\<bar> \<le> e > (\<exists>z. \<bar>zx\<bar> \<le> d & f z = y)" 

1352 
proof  

1353 
have "xd \<le> x+d" "\<forall>z. xd \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z" using cont d 

22998  1354 
by (auto simp add: abs_le_iff) 
21164  1355 
from isCont_Lb_Ub [OF this] 
1356 
obtain L M 

1357 
where all1 [rule_format]: "\<forall>z. xd \<le> z \<and> z \<le> x+d \<longrightarrow> L \<le> f z \<and> f z \<le> M" 

1358 
and all2 [rule_format]: 

1359 
"\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>z. xd \<le> z \<and> z \<le> x+d \<and> f z = y)" 

1360 
by auto 

1361 
with d have "L \<le> f x & f x \<le> M" by simp 

1362 
moreover have "L \<noteq> f x" 

1363 
proof  

1364 
from lemma_isCont_inj2 [OF d inj cont] 

1365 
obtain u where "\<bar>u  x\<bar> \<le> d" "f u < f x" by auto 

1366 
thus ?thesis using all1 [of u] by arith 

1367 
qed 

1368 
moreover have "f x \<noteq> M" 

1369 
proof  

1370 
from lemma_isCont_inj [OF d inj cont] 

1371 
obtain u where "\<bar>u  x\<bar> \<le> d" "f x < f u" by auto 

1372 
thus ?thesis using all1 [of u] by arith 

1373 
qed 

1374 
ultimately have "L < f x & f x < M" by arith 

1375 
hence "0 < f x  L" "0 < M  f x" by arith+ 

1376 
from real_lbound_gt_zero [OF this] 

1377 
obtain e where e: "0 < e" "e < f x  L" "e < M  f x" by auto 

1378 
thus ?thesis 

1379 
proof (intro exI conjI) 

23441  1380 
show "0<e" using e(1) . 
21164  1381 
show "\<forall>y. \<bar>y  f x\<bar> \<le> e \<longrightarrow> (\<exists>z. \<bar>z  x\<bar> \<le> d \<and> f z = y)" 
1382 
proof (intro strip) 

1383 
fix y::real 

1384 
assume "\<bar>y  f x\<bar> \<le> e" 

1385 
with e have "L \<le> y \<and> y \<le> M" by arith 

1386 
from all2 [OF this] 

1387 
obtain z where "x  d \<le> z" "z \<le> x + d" "f z = y" by blast 

27668  1388 
thus "\<exists>z. \<bar>z  x\<bar> \<le> d \<and> f z = y" 
22998  1389 
by (force simp add: abs_le_iff) 
21164  1390 
qed 
1391 
qed 

1392 
qed 

1393 
